# Example: Dietary fat

    library(multinma)
    options(mc.cores = parallel::detectCores())

This vignette describes the analysis of 10 trials comparing reduced fat diets to control (non-reduced fat diets) for preventing mortality (Hooper et al. 2000; Dias et al. 2011). The data are available in this package as `dietary_fat`:

    head(dietary_fat)
    #>   studyn            studyc trtn        trtc   r    n      E
    #> 1      1              DART    1     Control 113 1015 1917.0
    #> 2      1              DART    2 Reduced Fat 111 1018 1925.0
    #> 3      2 London Corn/Olive    1     Control   1   26   43.6
    #> 4      2 London Corn/Olive    2 Reduced Fat   5   28   41.3
    #> 5      2 London Corn/Olive    2 Reduced Fat   3   26   38.0
    #> 6      3    London Low Fat    1     Control  24  129  393.5

### Setting up the network

We begin by setting up the network - here just a pairwise meta-analysis. We have arm-level rate data giving the number of deaths (`r`) and the person-years at risk (`E`) in each arm, so we use the function `set_agd_arm()`. We set “Control” as the reference treatment.

    diet_net <- set_agd_arm(dietary_fat,
                            study = studyc,
                            trt = trtc,
                            r = r,
                            E = E,
                            trt_ref = "Control",
                            sample_size = n)
    diet_net
    #> A network with 10 AgD studies (arm-based).
    #>
    #> ------------------------------------------------------- AgD studies (arm-based) ----
    #>  Study                   Treatments
    #>  DART                    2: Control | Reduced Fat
    #>  London Corn/Olive       2: Control | Reduced Fat
    #>  London Low Fat          2: Control | Reduced Fat
    #>  Minnesota Coronary      2: Control | Reduced Fat
    #>  MRC Soya                2: Control | Reduced Fat
    #>  Oslo Diet-Heart         2: Control | Reduced Fat
    #>  STARS                   2: Control | Reduced Fat
    #>  Sydney Diet-Heart       2: Control | Reduced Fat
    #>  Veterans Administration 2: Control | Reduced Fat
    #>  Veterans Diet & Skin CA 2: Control | Reduced Fat
    #>
    #>  Outcome type: rate
    #> ------------------------------------------------------------------------------------
    #> Total number of treatments: 2
    #> Total number of studies: 10
    #> Reference treatment is: Control
    #> Network is connected

We also specify the optional `sample_size` argument, although it is not strictly necessary here. In this case `sample_size` would only be required to produce a network plot with nodes weighted by sample size, and a network plot is not particularly informative for a meta-analysis of only two treatments. (The `sample_size` argument is more important when a regression model is specified, since it also enables automatic centering of predictors and production of predictions for studies in the network, see `?set_agd_arm`.)

### Meta-analysis models

We fit both fixed effect (FE) and random effects (RE) models.

#### Fixed effect meta-analysis

First, we fit a fixed effect model using the `nma()` function with `trt_effects = "fixed"`. We use \(\mathrm{N}(0, 100^2)\) prior distributions for the treatment effects \(d_k\) and study-specific intercepts \(\mu_j\). We can examine the range of parameter values implied by these prior distributions with the `summary()` method:

    summary(normal(scale = 100))
    #> A Normal prior distribution: location = 0, scale = 100.
    #> 50% of the prior density lies between -67.45 and 67.45.
    #> 95% of the prior density lies between -196 and 196.

The model is fitted using the `nma()` function. By default, this will use a Poisson likelihood with a log link function, auto-detected from the data.

    diet_fit_FE <- nma(diet_net,
                       trt_effects = "fixed",
                       prior_intercept = normal(scale = 100),
                       prior_trt = normal(scale = 100))

Basic parameter summaries are given by the `print()` method:

    diet_fit_FE
    #> A fixed effects NMA with a poisson likelihood (log link).
    #> Inference for Stan model: poisson.
    #> 4 chains, each with iter=2000; warmup=1000; thin=1;
    #> post-warmup draws per chain=1000, total post-warmup draws=4000.
    #>
    #>                   mean se_mean   sd    2.5%     25%     50%     75%   97.5% n_eff Rhat
    #> d[Reduced Fat]   -0.01    0.00 0.05   -0.11   -0.04   -0.01    0.03    0.10  3659    1
    #> lp__           5386.21    0.06 2.35 5380.81 5384.82 5386.53 5387.93 5389.91  1556    1
    #>
    #> Samples were drawn using NUTS(diag_e) at Wed Nov 25 17:57:02 2020.
    #> For each parameter, n_eff is a crude measure of effective sample size,
    #> and Rhat is the potential scale reduction factor on split chains (at
    #> convergence, Rhat=1).

By default, summaries of the study-specific intercepts \(\mu_j\) are hidden, but could be examined by changing the `pars` argument:

    # Not run
    print(diet_fit_FE, pars = c("d", "mu"))

The prior and posterior distributions can be compared visually using the `plot_prior_posterior()` function:

    plot_prior_posterior(diet_fit_FE)

#### Random effects meta-analysis

We now fit a random effects model using the `nma()` function with `trt_effects = "random"`. Again, we use \(\mathrm{N}(0, 100^2)\) prior distributions for the treatment effects \(d_k\) and study-specific intercepts \(\mu_j\), and we additionally use a \(\textrm{half-N}(5^2)\) prior for the heterogeneity standard deviation \(\tau\). We can examine the range of parameter values implied by these prior distributions with the `summary()` method:

    summary(normal(scale = 100))
    #> A Normal prior distribution: location = 0, scale = 100.
    #> 50% of the prior density lies between -67.45 and 67.45.
    #> 95% of the prior density lies between -196 and 196.
    summary(half_normal(scale = 5))
    #> A half-Normal prior distribution: location = 0, scale = 5.
    #> 50% of the prior density lies between 0 and 3.37.
    #> 95% of the prior density lies between 0 and 9.8.

Fitting the RE model

    diet_fit_RE <- nma(diet_net,
                       trt_effects = "random",
                       prior_intercept = normal(scale = 10),
                       prior_trt = normal(scale = 10),
                       prior_het = half_normal(scale = 5))

Basic parameter summaries are given by the `print()` method:

    diet_fit_RE
    #> A random effects NMA with a poisson likelihood (log link).
    #> Inference for Stan model: poisson.
    #> 4 chains, each with iter=2000; warmup=1000; thin=1;
    #> post-warmup draws per chain=1000, total post-warmup draws=4000.
    #>
    #>                   mean se_mean   sd    2.5%     25%     50%     75%   97.5% n_eff Rhat
    #> d[Reduced Fat]   -0.02    0.00 0.09   -0.20   -0.06   -0.01    0.03    0.16  1693 1.00
    #> lp__           5378.68    0.12 3.90 5370.15 5376.24 5378.97 5381.40 5385.57  1109 1.00
    #> tau               0.13    0.00 0.11    0.00    0.05    0.10    0.18    0.41   923 1.01
    #>
    #> Samples were drawn using NUTS(diag_e) at Wed Nov 25 17:57:20 2020.
    #> For each parameter, n_eff is a crude measure of effective sample size,
    #> and Rhat is the potential scale reduction factor on split chains (at
    #> convergence, Rhat=1).

By default, summaries of the study-specific intercepts \(\mu_j\) and study-specific relative effects \(\delta_{jk}\) are hidden, but could be examined by changing the `pars` argument:

    # Not run
    print(diet_fit_RE, pars = c("d", "mu", "delta"))

The prior and posterior distributions can be compared visually using the `plot_prior_posterior()` function:

    plot_prior_posterior(diet_fit_RE, prior = c("trt", "het"))

#### Model comparison

Model fit can be checked using the `dic()` function:

    (dic_FE <- dic(diet_fit_FE))
    #> Residual deviance: 22.4 (on 21 data points)
    #>                pD: 11.2
    #>               DIC: 33.6

    (dic_RE <- dic(diet_fit_RE))
    #> Residual deviance: 21.3 (on 21 data points)
    #>                pD: 13.5
    #>               DIC: 34.8

Both models appear to fit the data well, as the residual deviance is close to the number of data points. The DIC is very similar between models, so the FE model may be preferred for parsimony.

We can also examine the residual deviance contributions with the corresponding `plot()` method.

    plot(dic_FE)

    plot(dic_RE)

### Further results

Dias et al. (2011) produce absolute predictions of the mortality rates on reduced fat and control diets, assuming a Normal distribution on the baseline log rate of mortality with mean \(-3\) and precision \(1.77\). We can replicate these results using the `predict()` method. The `baseline` argument takes a `distr()` distribution object, with which we specify the corresponding Normal distribution. We set `type = "response"` to produce predicted rates (`type = "link"` would produce predicted log rates).

    pred_FE <- predict(diet_fit_FE,
                       baseline = distr(qnorm, mean = -3, sd = 1.77^-0.5),
                       type = "response")
    pred_FE
    #>                   mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[Control]     0.07 0.06 0.01 0.03 0.05 0.08  0.22     3843     3814    1
    #> pred[Reduced Fat] 0.07 0.06 0.01 0.03 0.05 0.08  0.22     3824     3776    1
    plot(pred_FE)

    pred_RE <- predict(diet_fit_RE,
                       baseline = distr(qnorm, mean = -3, sd = 1.77^-0.5),
                       type = "response")
    pred_RE
    #>                   mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[Control]     0.07 0.06 0.01 0.03 0.05 0.08  0.21     3762     3706    1
    #> pred[Reduced Fat] 0.07 0.06 0.01 0.03 0.05 0.08  0.22     3781     3857    1
    plot(pred_RE)

If the `baseline` argument is omitted, predicted rates will be produced for every study in the network based on their estimated baseline log rate \(\mu_j\):

    pred_FE_studies <- predict(diet_fit_FE, type = "response")
    pred_FE_studies
    #> ------------------------------------------------------------------- Study: DART ----
    #>
    #>                         mean sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[DART: Control]     0.06  0 0.05 0.06 0.06 0.06  0.07     6024     3391    1
    #> pred[DART: Reduced Fat] 0.06  0 0.05 0.06 0.06 0.06  0.07     8793     3587    1
    #>
    #> ------------------------------------------------------ Study: London Corn/Olive ----
    #>
    #>                                      mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[London Corn/Olive: Control]     0.07 0.02 0.03 0.06 0.07 0.09  0.13     8936     3180    1
    #> pred[London Corn/Olive: Reduced Fat] 0.07 0.02 0.03 0.05 0.07 0.09  0.13     9289     3178    1
    #>
    #> --------------------------------------------------------- Study: London Low Fat ----
    #>
    #>                                   mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[London Low Fat: Control]     0.06 0.01 0.04 0.05 0.06 0.06  0.08     7776     3239    1
    #> pred[London Low Fat: Reduced Fat] 0.06 0.01 0.04 0.05 0.06 0.06  0.08     8878     3243    1
    #>
    #> ----------------------------------------------------- Study: Minnesota Coronary ----
    #>
    #>                                       mean sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[Minnesota Coronary: Control]     0.05  0 0.05 0.05 0.05 0.06  0.06     5490     3384    1
    #> pred[Minnesota Coronary: Reduced Fat] 0.05  0 0.05 0.05 0.05 0.06  0.06     7269     3471    1
    #>
    #> --------------------------------------------------------------- Study: MRC Soya ----
    #>
    #>                             mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[MRC Soya: Control]     0.04 0.01 0.03 0.04 0.04 0.04  0.05     7438     2610    1
    #> pred[MRC Soya: Reduced Fat] 0.04 0.01 0.03 0.04 0.04 0.04  0.05     8139     2753    1
    #>
    #> -------------------------------------------------------- Study: Oslo Diet-Heart ----
    #>
    #>                                    mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[Oslo Diet-Heart: Control]     0.06 0.01 0.05 0.06 0.06 0.07  0.08     5747     3111    1
    #> pred[Oslo Diet-Heart: Reduced Fat] 0.06 0.01 0.05 0.06 0.06 0.07  0.08     7256     2867    1
    #>
    #> ------------------------------------------------------------------ Study: STARS ----
    #>
    #>                          mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[STARS: Control]     0.02 0.01 0.01 0.01 0.02 0.03  0.05     7661     2450    1
    #> pred[STARS: Reduced Fat] 0.02 0.01 0.01 0.01 0.02 0.03  0.05     7599     2461    1
    #>
    #> ------------------------------------------------------ Study: Sydney Diet-Heart ----
    #>
    #>                                      mean sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[Sydney Diet-Heart: Control]     0.03  0 0.03 0.03 0.03 0.04  0.04     6249     2576    1
    #> pred[Sydney Diet-Heart: Reduced Fat] 0.03  0 0.03 0.03 0.03 0.04  0.04     6924     3169    1
    #>
    #> ------------------------------------------------ Study: Veterans Administration ----
    #>
    #>                                            mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS
    #> pred[Veterans Administration: Control]     0.11 0.01  0.1 0.11 0.11 0.12  0.13     5690     3222
    #> pred[Veterans Administration: Reduced Fat] 0.11 0.01  0.1 0.11 0.11 0.12  0.13     7106     3011
    #>                                            Rhat
    #> pred[Veterans Administration: Control]        1
    #> pred[Veterans Administration: Reduced Fat]    1
    #>
    #> ------------------------------------------------ Study: Veterans Diet & Skin CA ----
    #>
    #>                                            mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS
    #> pred[Veterans Diet & Skin CA: Control]     0.01 0.01    0 0.01 0.01 0.02  0.03     6097     2463
    #> pred[Veterans Diet & Skin CA: Reduced Fat] 0.01 0.01    0 0.01 0.01 0.02  0.03     6099     2416
    #>                                            Rhat
    #> pred[Veterans Diet & Skin CA: Control]        1
    #> pred[Veterans Diet & Skin CA: Reduced Fat]    1
    plot(pred_FE_studies) + ggplot2::facet_grid(Study~., labeller = ggplot2::label_wrap_gen(width = 10))

## References

Dias, S., N. J. Welton, A. J. Sutton, and A. E. Ades. 2011. “NICE DSU Technical Support Document 2: A Generalised Linear Modelling Framework for Pair-Wise and Network Meta-Analysis of Randomised Controlled Trials.” National Institute for Health and Care Excellence. http://nicedsu.org.uk/.

Hooper, L., C. D. Summerbell, J. P. T. Higgins, R. L. Thompson, G. Clements, N. Capps, G. Davey Smith, R. Riemersma, and S. Ebrahim. 2000. “Reduced or Modified Dietary Fat for Preventing Cardiovascular Disease.” Cochrane Database of Systematic Reviews, no. 2. https://doi.org/10.1002/14651858.CD002137.